# Exterior pairs and up step statistics on Dyck paths

Sen-Peng Eu, Tung Shan Fu

Research output: Contribution to journalArticle

### Abstract

Let Cn be the set of Dyck paths of length n. In this paper, by a new auto- morphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set Cn is equidistributed with the statistic 'number of up steps at height h with h ≡ 0 (mod 3)'. Moreover, for m ≥ 3, we prove that the two statistics 'number of up steps at height h with h ≡ 0 (mod m)' and 'number of up steps at height h with h ≡ m - 1 (mod m)' on the set Cn are 'almost equidistributed'. Both results are proved combinatorially.

Original language English Electronic Journal of Combinatorics 18 1 Published - 2011 May 12

Dyck Paths
Statistics
Statistic
Ordered Trees
Automorphism

### Keywords

• Continued fraction
• Dyck path
• Exterior pair
• Ordered tree
• Planted tree

### ASJC Scopus subject areas

• Theoretical Computer Science
• Geometry and Topology
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics
• Applied Mathematics

### Cite this

Exterior pairs and up step statistics on Dyck paths. / Eu, Sen-Peng; Fu, Tung Shan.

In: Electronic Journal of Combinatorics, Vol. 18, No. 1, 12.05.2011.

Research output: Contribution to journalArticle

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